Always have a contract. I have never performed children shows but I wouldn't show up without a signed contract.
You risk showing up, and being turned away by the parents because it slipped someone's mind to contact you that their plans changed.
Perhaps they ended up going with an entertainer that undercut your price, or went with a different type of entertainer.
Perhaps the kid came down with a flu or chickpox and they had to cancel the party.
And most importantly, there will be no misunderstandings on price.
Treat this as a business, and not a hobby.
And Fallingblood makes a great point. Not just so you don't end up as a babysitter. Have that contract to have an adult present, and also a limitation of liability (in addition to your performers' insurance). So you don't become the victim of a lawsuit. There will be no innapropriate touching claims, and being the only legal adult present in a room full of minors makes you liable and responsible for anything that happens in that room. Child choking, child slips and hits head, cuts themselves on something, pulls a heavy object over on themselves, etc. Have an adult that will take responsibility for the underage guests present at all times.
Anything can happen, and I'm sure you don't want the headaches involved if something does happen.
Sure it's paranoia at it's best, but it will protect you.